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What is the expected value for this
The exponent would not change. For example, take random points in a circular disk. Fit a triangle in the disk. A constant fraction of the n points falls inside the triangle (w.h.p.), and thus you get at least $c′n^{1/3}$ points in convex position; similarly, an ellipse (affine image of a disk) fits into any triangle, showing the other direction of the inequality. Any two convex shapes are related like this (after affine transformation).
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The Cayley Menger Theorem and integer matrices with row sum 2
corrected info about the sign
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The Cayley Menger Theorem and integer matrices with row sum 2
Yes, it is. I have updated my answer.
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The Cayley Menger Theorem and integer matrices with row sum 2
I wonder how a matrix with positive entries can have trace 0?
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functions satisfying "one-one iff onto"
typos, grammar
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long enough interval of integers to solve a simultaneous congruence
@Noam. Clarification: I was not puzzled because I did not see that the product has "distinct" exponents, after expanding the product and before collecting terms with the same exponent. But why is this fact needed?
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long enough interval of integers to solve a simultaneous congruence
Nice! I am puzzled by the remark that the $N$ monomials have distinct exponents. If they were not distinct, just collect terms and let them cancel if they want, and let $m_j$ only refer to the nonzero terms. This would reduce the number of terms of $P(X)$, and the argument goes through (with a reduced number $N$) as long as the polynomial does not become identically zero. But this is ok, as we know that the constant term is nonzero because it has abs. value 1. (And anyway, who is afraid of the Combinatorial Nullstellensatz? It doesn't even need such advanced stuff as "complex numbers;-)
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German mathematical terms like "Nullstellensatz"
There is also the inverse tendency that the German terms tend to be forgotten, now that English has become so prevalent. Many German students will happily use "Konvolution" when they read it in a paper before I teach them to use "Faltung". Similarly, "bottleneck" like in "bottleneck objective function" tends to be sometimes literally translated into "Flaschenhals" instead of "Engpass" (meaning narrow pass, which is (or used to be) the usual term in this situation). A case which I particularly deplore is the thoughtless translation of "line segment" into "Liniensegment" instead of "Strecke".
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How does changing the transition probabilities affect the concentration of a position-dependent random walk?
The question did not ask for the existence of a speed originally. Maybe $X(t)$ is concentrated around $f(t)$ for some function $f$. If we assume that $f(t)$ increases unboundedly, is there an example where the random walk is concentrated, but the modified random walk is not concentrated (around a different function $f'(t)$)?
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How does changing the transition probabilities affect the concentration of a position-dependent random walk?
Should the assumption hold for all $c$? Is it essential that the walk moves to the right on average (like $(1/3)t$ in your example) our would a movement to the left or a stationary concentration around $0$ be permitted? (In the last cases there would probably be easy counterexamples when some $p_n$ is changed from 0 to $\epsilon$.)
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Is this min not less than a min
@userior, why don't you calculate the value for the square plus center and compare? Then at least we have a conjecture, and we only need to concentrate on one problem.
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Proofs without words
2nd proof: It would be nicer if the small strips were above and to the left of the big square.
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Untangling entwined rigid chains in 3-space
Anyway, When you specify the steps in this form, another important 3-dimensional class of motions is excluded: screw motions (rotation around an axis with simultaneous translation along that axis). Of course such a motion can be approximated by a sequence of translations and rotations, but the number of steps might be large (not bounded in $n$). I imagine two chains that require a screw motion at one point. It would be analogous to allowing a point in the plane to move only vertically or horizontally. The necessary number of such steps inside a slanted strip can be as large as we like.
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Untangling entwined rigid chains in 3-space
"rotation about a fixed point" is maybe inappropriate if 3d. It might be rotation about a fixed axis.
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Is this min not less than a min
@partial results: A natural conjecture is that the second quantity is minimized when the 5 points form a regular pentagon. Have you checked this? What is the value for a regular pentagon? Is the first quantity perhaps minimized for a square plus center? What is the value for this configuration? What is the best value that you know for each of these quantities?